Finite derivation type implies the homological finiteness condition \(FP_ 3\)
type: Item
instance of: scholarly article
title: Finite derivation type implies the homological finiteness condition \(FP_ 3\)
zbMATH Open document ID: 0834.20065
author: Robert Cremanns
author: Friedrich Otto
published in: Journal of Symbolic Computation
publication date: 1996-03-18T00:00:00Z
review text: Let \(M\) be a finitely generated monoid. A monoid presentation will consist of a finite alphabet \(\Sigma\) and a subset \(R\subseteq\Sigma^*\times\Sigma^*\) of rewriting rules (in the sense of formal grammar). If \(M\) is isomorphic to the monoid \(\Sigma^*\)/(the congruence generated by \(R\)), then the pair \((\Sigma;R)\) is called a presentation for \(M\). To say that a presentation \((\Sigma;R)\) is of finite derivation type means, roughly, that there is a finite set of derivations (in the sense of term rewriting) which, up to pre- and post-multiplication by elements of \(\Sigma^*\), account for all word equivalences in \(\Sigma^*\). Specifically, one first forms a graph \(\Gamma=\Gamma(\Sigma; R)\) with vertices \(V=\Sigma^*\) and, for \(u, v \in \Sigma^*\) and \((l,r) \in R\), edges \((u, (l, r), v, \pm 1)\) going from \(ulk\) to \(urv\) (in the case of \(+1\), and in the opposite direction in the case of \(-1\)). In this graph one can then consider the set of paths, \(P(\Gamma)\), and the equivalence relation \(P^{(2)}\) on these paths which has two paths \(p\) and \(q\) equivalent exactly when they have the same source and target vertices. Let \(B\) be a subset of \(P^{(2)}\) (considered as a set of pairs of paths). Then the homotopy relation generated by \(B\) on \(P(\Gamma)\) is the smallest relation \(\simeq\) which satisfies i. if \(p \simeq q\), then \(xpy \simeq xqy\) for all \(x, y \in \Sigma^*\), ii. if \(p \simeq q\) and \(r, s \in P(\Gamma)\), then \(r \circ p \circ s \simeq r \circ q \circ s\), provided the composition of paths is defined, and iii. \(D \cup I \cup B \subseteq \simeq\), where \(D\) represents the ``commuting squares'' resulting from applying two rewriting rules \((l_1, r_1)\) and \((l_2, r_2)\) in the two possible orders, and \(I\) represents the relation forcing a length 2 path consisting of traversing (essentially) the same edge ``out and back'' to be equivalent to the identity. If there is a finite \(B\) such that the homotopy…
Mathematics Subject Classification ID: 20M35
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 68Q42
Mathematics Subject Classification ID: 18G20
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 696963
zbMATH Keywords: finitely generated monoids
zbMATH Keywords: monoid presentations
zbMATH Keywords: rewriting rules
zbMATH Keywords: finite derivation type
zbMATH Keywords: monoid rings
zbMATH Keywords: homological finiteness condition \(FP_ 3\)
zbMATH Keywords: finitely generated groups
reviewed by: William R. Nico
MaRDI profile type: Publication
full work available at URL: jsco.1994.1039
OpenAlex ID: W1997476598
DOI: 10.1006/JSCO.1994.1039
description: scientific article; zbMATH DE number 696963